For more rigorous and precise analysis, the numerical approximations must be used.

In the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument.

These generalizations of the convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

* Conditioning ( numerical analysis ), a quantity describing if a numerical problem is well-behaved

In numerical analysis and functional analysis, a discrete wavelet transform ( DWT ) is any wavelet transform for which the wavelets are discretely sampled.

The Euler – Maclaurin formula is also used for detailed error analysis in numerical quadrature.

Data analysis encompasses methods for processing both quantitive ( numerical ) and qualitative ( non-numerical ) research data.

Originally developed by IBM at their campus in south San Jose, California in the 1950s for scientific and engineering applications, Fortran came to dominate this area of programming early on and has been in continual use for over half a century in computationally intensive areas such as numerical weather prediction, finite element analysis, computational fluid dynamics, computational physics and computational chemistry.

Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis and other areas.

He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory.

Elementary row operations are used to reduce a matrix to what is called triangular form ( in numerical analysis ) or row echelon form ( in abstract algebra ).

By the 1920s Lewis Fry Richardson's interest in weather prediction led him to propose human computers and numerical analysis to model the weather ; to this day, the most powerful computers on Earth are needed to adequately model its weather using the Navier – Stokes equations.

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

* In numerical analysis and in signal processing, a time series can be " integrated " numerically by various step-by-step means, including autoregressive integrated moving average and the Runge-Kutta methods.

Mathematicians do research in fields such as logic, set theory, category theory, abstract algebra, number theory, analysis, geometry, topology, dynamical systems, combinatorics, game theory, information theory, numerical analysis, optimization, computation, probability and statistics.

It was later found, through numerical analysis, that this was due to numerical instability.

It is now also used in education, in particular the teaching of linear algebra and numerical analysis, and is popular amongst scientists involved in image processing.

# Scientific computing and numerical analysis.

Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.

Iterative numerical methods, such as Newton's method or the three-body problem, are also algorithms which are inherently serial.

The seventeenth-century English physicist and mathematician, Isaac Newton, developed a wealth of new mathematics ( for example, calculus and several numerical methods such as Newton's method to solve problems in physics.

As for a numerical aspect, it is known that Newton's method is delicate when finding the roots of the secular equation.

In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method.

Verlet integration () is a numerical method used to integrate Newton's equations of motion

# Via numerical optimization such as the conjugate gradient method or Newton's method.

The instrumental method, however, is about 100 times more sensitive and yields numerical results which can be accurately repeated at will over a period of time.

Chan's method for estimating the mean is numerically unstable when and both are large, because the numerical error in is not scaled down in the way that it is in the case.

As a general rule of thumb, if the condition number, then you may lose up to digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods.

* Conjugate gradient method, an algorithm for the numerical solution of particular systems of linear equations

Degen asked Abel to give a numerical example of his method and, while trying to provide an example, Abel discovered a mistake in his paper.

This method has become quite rare in modern times, having been largely superseded by cladistic analyses, as numerical taxonomy is sensitive to being misled by plesiomorphic traits.

However he also mentioned in the same report: " Meanwhile attention is being turned to the still difficult, but less unpromising, problem of radio detection and numerical considerations on the method of detection by reflected radio waves will be submitted when required.

The path integral Monte Carlo method is a numerical approach for determining the values of free energies, based on quantum dynamical principles.

A variation of the cutback technique is the substitution method, in which measurements are made on a full length of fiber, and then on a short length of fiber having the same characteristics ( core size, numerical aperture ), with the results from the short length being subtracted to give the results for the full length.

Before the advent of VLSI devices, TTL integrated circuits were a standard method of construction for the processors of mini-computer and mainframe processors ; such as the DEC VAX and Data General Eclipse, and for equipment such as machine tool numerical controls, printers and video display terminals.

In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved ; this is known as the method of generating functions.

A discrete element method ( DEM ), also called a distinct element method is any of family of numerical methods for computing the motion and effect of a large number of small particles.

The components are regarded as random variables, and may be grouped into two categories according to the method used to estimate their numerical values:

A common method is Platt's Sequential Minimal Optimization ( SMO ) algorithm, which breaks the problem down into 2-dimensional sub-problems that may be solved analytically, eliminating the need for a numerical optimization algorithm.

It was largely by this method ( mixed-race Bermudians being added to the number of Blacks, rather than added to the number of Whites or being defined as a separate demographic group ) that Coloured ( subsequently redefined in the twentieth century as Black ) Bermudians came to outnumber White Bermudians by the end of the 19th century, despite starting off at a numerical disadvantage, and despite low Black immigration prior to the latter 19th century ( other contributing factors included the scale of White relative to Black emigration in the 17th and 18th centuries, the greater mortality of Whites from disease in the late 17th century, and large-scale West Indian immigration, which began, like Portuguese immigration, in the 19th century to provide labourers for the new export agriculture industry and expansion of the Royal Naval Dockyard.

In mathematics, particularly linear algebra and numerical analysis, the Gram – Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space R < sup > n </ sup >.

Although the solutions cannot always be expressed exactly with radicals, they can be computed to any desired degree of accuracy using numerical methods such as the Newton – Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees.